Computing Homotopy Types of Directed Flag Complexes

Abstract

Combinatorially and stochastically defined simplicial complexes often have the homotopy type of a wedge of spheres. A prominent conjecture of Kahle quantifies this precisely for the case of random flag complexes. We explore whether such properties might extend to graphs arising from nature. We consider the brain network (as reconstructed by Varshney & al.) of the Caenorhabditis elegans nematode, an important model organism in biology. Using an iterative computational procedure based on elementary methods of algebraic topology, namely homology, simplicial collapses and coning operations, we show that its directed flag complex is homotopy equivalent to a wedge of spheres, completely determining, for the first time, the homotopy type of a flag complex corresponding to a brain network.We also consider the corresponding flag tournaplex and show that torsion can be found in the homology of its local directionality filtration. As a toy example, directed flag complexes of tournaments from McKay's collection are classified up to homotopy. Moore spaces other than spheres occur in this classification. As a tool, we prove that the fundamental group of the directed flag complex of any tournament is free by considering its cell structure.

Simplicial $q$-connectivity of directed graphs with applications to network analysis

“…A particular feature of the network is an overrepresentation of reciprocally connected triangle motifs and an underrepresentation of directed triangle cycles. The synaptic connectivity is obtainable from [1] and we used the steps in [22] to construct the directed graph. The graph we used has 279 vertices and 2,194 edges.…”

“…Our main example of a topological space on a digraph G is the directed flag complex, which is constructed from the directed cliques of G. For example, a 2-simplex is given by an ordered sequence of vertices (v 0 , v 1 , v 2 ) whenever any ordered pair (v i , v j ), for i < j, is a directed edge in G. By construction the simplices are endowed with an inherent directionality. For a recent work on computing the homotopy type of the directed flag complex of the C. elegans neuronal network, see [22].…”